Saturday, July 18, 2009

Now, Judge, I Had Debts No Honest Man Could Pay/The Bank Was Holding My Mortage And They Were Gonna Take My House Away

1.0 IntroductionThis example illustrates one aspect of how Sraffa analyzed natural resources. In this case, natural resources consist of land of various qualities or grades. The quantity of each grade is given; more land cannot be produced. Land is not destroyed either. Appropriate production processes yield as much land as a joint output as given as input. So this sort of model does not incorporate a natural resource like oil that is used up in production.

This example demonstrates that owners of less efficient (productive) natural resources can receive a greater rent.

By the way, Sraffa introduces a distinction between basic and non-basic commodities. Lands with positive rent are non-basic, and therefore their rent is a candidate for taxation.

2.0 TechnologyThis is a simple economy in which only corn is produced. Table 1 shows the available processes that have corn as an output. Each process requires the use of one grade of land, and no more than one process is known for each grade of land. (This is an example of a model of extensive rent.) Suppose this economy has available 150 acres of land of grade I, 162 acres of grade II land, and 210 acres of grade III land.

Table 1: Technology

Inputs

Process

Alpha

Beta

Gamma

Labor (Person-Years)

2/5

1

2/7

Grade I Land (Acres)

1

0

0

Grade II Land (Acres)

0

3/2

0

Grade III Land (Acres)

0

0

1

Corn (Bushels)

2/5

1/6

4/7

Output (Bushels)

1

1

1

3.0 PricesThe question to be addressed is what prices and distribution of income are compatible with a long-period position, given the technology. The amount of corn required for net output is a parameter that must be known to answer this question.

3.1 When Only One Grade Of Land Is CultivatedSuppose the requirements for use for corn in this economy can be satisfied by cultivating any one of the three grades of land. Two grades, and maybe some of the third grade, can lie fallow. So at most 90 bushels are produced each year, after replacinging up the seed corn.

And suppose that the wage is 1/2 bushels per person-year. The wage is paid out at the end of the year. These assumptions are enough to derive the factor-price curves shown in Figure 1. These curves are drawn under the assumption that all grades of land paid no rent.

Figure 2: Factor Price Curves

Since the beta factor-price curve is rightmost (on the outer frontier) at the given wage, all corn is produced on land of the second grade, and this land pays no rent. The wage and the rate of profits must satisfy the following equation:

(1/6)(1 + r) + w = 1

where a bushel corn is the numeraire. For a wage of 1/2 bushels per person-year, the rate of profits is 200%.

Under the given assumptions, the cost of producing a bushel corn on the first grade of land, even if that land were to pay no rent, is 7/5 bushels. Since this cost exceeds the revenues from selling a bushel of land, no capitalist would want to produce on the first grade of land. The reader can check that the cost of producing a bushel corn on the third grade of land also exceeds unity.

3.2 When Two Grades Of Land Are CultivatedNow suppose the requirements for use are such that two grades of land must be cultivated. The net output of this economy is between 90 and 180 bushels corn. The wage remains 1/2 bushels per person-year. In this case, the first and second grades are cultivated, with the second grade paying rent. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ1 = 1

(1/6)(1 + r) + w + (3/2)ρ2 = 1

ρ1ρ2 = 0

ρ1, ρ2 ≥ 0

The equations specify that no land can have a negative rent and that at least one grade of land must have a rent of zero. The rate of profits is 100%, when the wage is 1/2 bushels per person-year. Land of grade I pays no rent, and the rent on land of grade II is 1/9 bushels per acre.

The cost of producing a bushel corn on the third grade of land, accounting just for outlays of seed corn and labor, is 9/7 bushels. Capitalists will not want to cultivate the third grade of land, even if it is free.

3.3 When Three Grades Of Land Are CultivatedFinally, suppose the requirements for use for use require all three grades of land to be cultivated. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ1 = 1

(1/6)(1 + r) + w + (3/2)ρ2 = 1

(4/7)(1 + r) + (2/7)w + ρ3 = 1

ρ1ρ2, ρ3 = 0

ρ1, ρ2, ρ3 ≥ 0

The rate of profits is 50%, when the wage is 1/2 bushels per person-year. The rent on land of grade I is 1/5 bushels per acre. The rent on grade II land is 1/6 bushels per acre

3.4 Orders Of Efficiency And RentabilityThe above analysis has identified a definite order in which different grades of land will be cultivated as greater quantities of output are required to be produced. This is the order of efficiency. In this example, the order of efficiency, from most efficient to least efficient, is: Grade II, Grade I, Grade III.

One can also rank the grades of land from high rent to low rent, when all three grades of land are cultivated and the wage is 1/2 bushels per person-year. The order of rentability, from highest rent to zero rent, is: Grade I, Grade II, Grade III.

The orders of rentability and efficiency differ. It is possible for a less productive, that is, the less efficient, resource to provide its owner with a greater rent than the more efficient resource.

This example is not driven by the existence of switch points.

Reference

Alberto Quadrio-Curzio, "Rent, Income Distribution, and Orders of Efficiency and Rentability", in Essays on the Theory of Joint Production (Edited by L. L. Pasinetti), Columbia University Press, 1980.